Finite Group Databasebeta

← #27372040

Subgroups — #27372040

Order 13568.

922 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285836C211yesmaximal, characteristic, complemented
2#20285630C211yesmaximal, characteristic, complemented
3#20285290C211yesmaximal, characteristic, complemented
4#20285167C211yesmaximal, characteristic, complemented
5C2×#16810971C211yesmaximal, characteristic, complemented
6#20274387C212yesmaximal, complemented
7#20274379C212yesmaximal, complemented
8#20274322C212yesmaximal, complemented
9#20274305C212yesmaximal, complemented
10M16×#88322C211yesmaximal, characteristic, complemented
11C53×#3152C211yesmaximal, characteristic, complemented
12C2×#13133245—22no—
13#16811173C2211yescharacteristic
14C2×#13133214C2211yescharacteristic
15#16811026C2211yescharacteristic, complemented
16#16811026C2212yescomplemented
17#16811012—22no—
18#16811000C2212yes—
19#16810999C2212yes—
20#16810999C2211yescharacteristic, complemented
21#16810996C2212yes—
22#16810971—22no—
23#16810971C2212yes—
24#16810971C2212yes—
25#16810971C2212yes—
26#16810971C2211yescharacteristic, complemented
27#16810971C2212yescomplemented
28#16810969C2211yescharacteristic, complemented
29#16810936—22no—
30#16810900—22no—
31#16810857—22no—
32#16810818—22no—
33#16810756—22no—
34C4×#11719922—22no—
35#16810676C2211yescharacteristic
36C2×#13133137—22no—
37C2×C4×#88322C2211yescharacteristic
38#16810344C2212yescomplemented
39C8×#88322C2212yescomplemented
40C2×#137×C53—22no—
41C53×#532C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#497—22no—
44C53×#494C2212yescomplemented
45C53×#492C2212yescomplemented
46C4×M16×C53C2211yescharacteristic
47C2×#11719945—21no—
48C22×D212—21no—
49C22×D212—21no—
50C22×D212—21no—
51#13133245—48no—
52C2×#11719913—24no—
53C2×D424—24no—
54C2×#11719922—22no—
55C2×#11719922—24no—
56C22×#88322D412yes—
57#13133217C2311yescharacteristic
58#13133214—21no—
59#13133214—21no—
60#13133214—21no—
61#13133214C2312yes—
62#13133214C2311yescharacteristic
63#13133205—21no—
64#13133191—21no—
65#13133191—21no—
66C4×D212—21no—
67C4×D212—21no—
68#13133150—22no—
69#13133150—24no—
70C4⋊#88322D412yes—
71#13133138—22no—
72#13133137—24no—
73#13133137—22no—
74#13133137—24no—
75#13133137—24no—
76#13133137—24no—
77C22⋊#88322D412yes—
78C4×#88322D412yes—
79C4×#88322C2311yescharacteristic
80C4×#88322C2312yes—
81C4×#88322C2311yescharacteristic
82C2×#41×C53—21no—
83C22×Q8×C53—21no—
84#137×C53—48no—
85C2×Q16×C53—24no—
86C2×SD16×C53—24no—
87C2×M16×C53D412yes—
88C4⋊Q8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53C2312yes—
92C4⋊Q8×C53C2311yescharacteristic
93#127×C53C2311yescharacteristic
94#124×C53—21no—
95#121×C53—21no—
96#121×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C42×C53C2311yescharacteristic
102#97×C53D412yes—
103C4×C8×C53C2312yes—
104C23×D53—41no—
105#41×C53—44no—
106C2×Q8×C53—21no—
107C2×Q8×C53—22no—
108C2×Q8×C53—41no—
109C2×Q8×C53—44no—
110C2×D4×C53—21no—
111C2×D4×C53—21no—
112C2×D4×C53—21no—
113C2×D4×C53—41no—
114C22×C4×C53—41no—
115C22×C4×C53—41no—
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C2×C22⋊D53—41no—
120#11719945—44no—
121C2×D212—21no—
122C2×D212—22no—
123C2×D212—21no—
124C2×D212—22no—
125C2×D212—21no—
126C2×D212—22no—
127C2×D212—44no—
128C2×D212—44no—
129C2×D212—41no—
130C2×D212—41no—
131C2×D212—41no—
132C2×D212—82no—
133C2×C4×D53—41no—
134C2×Dic106—21no—
135Q16×C53—48no—
136SD16×C53—48no—
137M16×C53D818yescomplemented
138C2×C8×C53C2×D414yes—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C22⋊C4×C53—41no—
147C42×C53C2×D411yescharacteristic
148C42×C53C2×D411yescharacteristic
149C42×C53C2×D411yescharacteristic
150C42×C53C2411yescharacteristic
151#11719922—44no—
152#11719922—84no—
153#11719921—41no—
154#11719921—41no—
155C4⋊Dic53—21no—
156C4⋊Dic53—21no—
157C2×#88322—42no—
158C2×#88322C2×D412yes—
159C2×#88322C2×D412yes—
160C2×#88322C2×D414yes—
161D424—48no—
162#11719913—48no—
163C23×C53—41no—
164C23×C53#12711yescharacteristic
165C22×D53—41no—
166C22×D53—42no—
167C22×D53—82no—
168C22×D53—82no—
169C22×D53—82no—
170Q8×C53—44no—
171Q8×C53—42no—
172Q8×C53—82no—
173D4×C53—42no—
174D4×C53—42no—
175D4×C53—42no—
176D4×C53—82no—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—41no—
180C2×C4×C53—42no—
181C2×C4×C53—82no—
182C2×C4×C53—82no—
183C2×C4×C53#12711yescharacteristic
184C2×C4×C53#12711yescharacteristic
185C2×C4×C53#12711yescharacteristic
186C2×C4×C53C2×D814yes—
187C2×C4×C53C22×D411yescharacteristic
188C2×C4×C53C22×D411yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C22⋊D53—82no—
191C2×Dic53—41no—
192D212—44no—
193D212—42no—
194D212—44no—
195D212—42no—
196D212—42no—
197D212—44no—
198D212—82no—
199D212—82no—
200D212—82no—
201C4×D53—82no—
202Dic106—42no—
203C8×C53C2×D818yes—
204#88322—22no—
205#88322—22no—
206#88322—24no—
207#19631—531nomaximal, Sylow
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—81no—
211C22×C53—81no—
212C22×C53—81no—
213C22×C53#49312yes—
214C22×C53C2×#12711yescharacteristic
215C2×D53—81no—
216C2×D53—82no—
217C2×D53—82no—
218C2×D53—81no—
219C2×D53—82no—
220C2×D53—81no—
221C2×D53—164no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—81no—
225C4×C53—82no—
226C4×C53#49312yes—
227C4×C53C22×D812yes—
228C4×C53#57512yes—
229C4×C53#57712yes—
230Dic53—81no—
231#3226—531no—
232#3205—531no—
233#3169—531no—
234#3152D5311yescharacteristic, complemented
235#3151—531no—
236C2×#486—531no—
237#1722—532no—
238#1721—532no—
239#1709—532no—
240#1706—532no—
241C8×M16—531no—
242C2×C53—22no—
243C2×C53—161no—
244C2×C53C2×#49311yescharacteristic
245C2×C53#315311yescharacteristic
246C2×C53#315411yescharacteristic
247D53—161no—
248D53—162no—
249C2×#137—22no—
250C2×AGL(1,ℤ/8)—1062no—
251#532—531no—
252#532C2×D5311yescharacteristic
253C2×C4⋊Q8C2×D5311yescharacteristic
254C2×#127—531no—
255#500—531no—
256#500—532no—
257#497—22no—
258#496—1062no—
259#494C2×D5312yescomplemented
260#493—531no—
261#493—532no—
262#493—532no—
263#492—532no—
264#492C2×D5312yescomplemented
265#486—532no—
266#486—532no—
267#486—531no—
268#486—532no—
269#486—532no—
270#486—531no—
271#486—1062no—
272#480—1062no—
273#474—1062no—
274#466—1062no—
275#459—1062no—
276#447—1062no—
277C4×D8—1062no—
278#431—531no—
279C2×#102—1062no—
280C4×M16C2×D5311yescharacteristic
281C2×C4×C8—531no—
282#322—532no—
283C82—532no—
284C53#1963111yescharacteristic, complemented, Sylow
285C2×#41—21no—
286C2×#41—1061no—
287C22×Q8—21no—
288C22×D4—1061no—
289C22×D4—1061no—
290C22×D4—1061no—
291#137—48no—
292AGL(1,ℤ/8)—2128no—
293C2×Q16—24no—
294C2×SD16—24no—
295C2×SD16—1064no—
296C2×D8—1064no—
297C2×D8—1062no—
298C2×D8—1064no—
299C2×M16C22⋊D5312yes—
300C22×C8—532no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—531no—
305C4⋊Q8C22×D5312yes—
306C4⋊Q8C22×D5311yescharacteristic
307#127—532no—
308#127—531no—
309#127—1061no—
310#127—1061no—
311#127—1061no—
312#127C22×D5311yescharacteristic
313#124—21no—
314#124—1061no—
315#121—21no—
316#121—21no—
317#121—1061no—
318#121—1061no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—1061no—
322C4×D4—1061no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C42C22×D5311yescharacteristic
326#107—1062no—
327#107—1064no—
328C4⋊C8—532no—
329#102—1064no—
330#102—1064no—
331#102—1062no—
332#102—1064no—
333#102—1064no—
334#102—1062no—
335C22⋊C8—532no—
336#97C22⋊D5312yes—
337C4×C8—532no—
338C4×C8—531no—
339C4×C8—532no—
340C4×C8—531no—
341C4×C8C22×D5312yes—
342C24—2121no—
343#41—44no—
344#41—2124no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—44no—
349C2×Q8—1061no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—1061no—
355C2×D4—1062no—
356C2×D4—1061no—
357C2×D4—1062no—
358C2×D4—1061no—
359C2×D4—1062no—
360C2×D4—4242no—
361C2×D4—2124no—
362C2×D4—2121no—
363C2×D4—2121no—
364C2×D4—2121no—
365C2×D4—2121no—
366C2×D4—2124no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—2121no—
370C22×C4D4×D5311yescharacteristic
371C22×C4D4×D5311yescharacteristic
372C22×C4C2×C22⋊D5311yescharacteristic
373Q16—48no—
374SD16—48no—
375SD16—2128no—
376D8—4244no—
377D8—2124no—
378D8—2128no—
379M16#1171992218yescomplemented
380C2×C8—532no—
381C2×C8—532no—
382C2×C8—534no—
383C2×C8—2122no—
384C2×C8C2×C22⋊D5314yes—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—1061no—
392C4⋊C4—1061no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—2121no—
396C22⋊C4—2121no—
397C42D4×D5311yescharacteristic
398C42D4×D5311yescharacteristic
399C42C2×C22⋊D5311yescharacteristic
400C42C23×D5311yescharacteristic
401C23—41no—
402C23—4242no—
403C23—4242no—
404C23—4242no—
405C23—2121no—
406C23—2122no—
407C23#1313321311yescharacteristic
408Q8—44no—
409Q8—42no—
410Q8—82no—
411Q8—2122no—
412D4—42no—
413D4—42no—
414D4—42no—
415D4—82no—
416D4—4242no—
417D4—4242no—
418D4—4242no—
419D4—4242no—
420D4—2122no—
421D4—2124no—
422D4—2122no—
423D4—2124no—
424D4—2124no—
425D4—2122no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—4242no—
433C2×C4—2121no—
434C2×C4#1313321311yescharacteristic
435C2×C4#1313321311yescharacteristic
436C2×C4#1313321311yescharacteristic
437C2×C4C2×#1171992214yes—
438C2×C4C2×D4×D5311yescharacteristic
439C2×C4C2×D4×D5311yescharacteristic
440C2×C4C22×C22⋊D5311yescharacteristic
441C8—1062no—
442C8—1062no—
443C8—1064no—
444C8C2×#1171992218yes—
445C22—21no—
446C22—21no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—8484no—
451C22—4242no—
452C22—4241no—
453C22—4241no—
454C22—4242no—
455C22—4242no—
456C22—4241no—
457C22#1681099912yes—
458C22C2×#1313321311yescharacteristic
459C4—21no—
460C4—21no—
461C4—81no—
462C4—82no—
463C4—4241no—
464C4#1681099912yes—
465C4C22×#1171992212yes—
466C4#1681142212yes—
467C4#1681143212yes—
468C2—22no—
469C2—161no—
470C2—8482no—
471C2—8481no—
472C2C2×#1681099911yescharacteristic
473C2#2028518211yescharacteristic
474C2#2028519411yescharacteristic
subgroupG/H
centerC22C2×#13133213
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3152C2
socleC22×C53C2×#127
hypercenter#3152D53
hypocenterC53#19631
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20285836C2semidirect
2. #20285630C2semidirect
3. #20285290C2semidirect
4. #20285167C2semidirect
5. C2×#16810971C2semidirect
6. #20274387C2semidirect
7. #20274379C2semidirect
8. #20274322C2semidirect
9. #20274305C2semidirect
10. M16×#88322C2semidirect
11. C53×#3152C2semidirect
15. #16811026C22semidirect
16. #16811026C22semidirect
20. #16810999C22semidirect
26. #16810971C22semidirect
27. #16810971C22semidirect
28. #16810969C22semidirect
38. #16810344C22semidirect
39. C8×#88322C22semidirect
44. C53×#494C22semidirect
45. C53×#492C22semidirect
137. M16×C53D8semidirect
234. #3152D53semidirect
259. #494C2×D53semidirect
264. #492C2×D53semidirect
284. C53#19631semidirect
379. M16#11719922semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202858362
#202856302
#202852902
#202851672
C2×#168109712
#202743872
#202743792
#202743222
#202743052
M16×#883222
C53×#31522
#1963153